This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course. This is a revised edition, new examples and exercises have been added, and many minor improvements have been made throughout the text. The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. Foreword v(4) Prerequisites ix PART ONE Basic Theory 1(290) CHAPTER I Complex Numbers and Functions 3(34) 1. Difinition 3(5) 2. Polar Form 8(4) 3. Complex Valued Functions 12(5) 4. Limits and Compact Sets 17(10) Compact Sets 21(6) 5. Complex Differentiability 27(4) 6. The Cauchy-Riemann Equations 31(2) 7. Angles Under Holomorphics Maps 33(4) CHAPTER II Power Series 37(49) 1. Formal Power Series 37(10) 2. Convergent Power Series 47(13) 3. Relations Between Formal and Convergent Series 60(8) Sums and Products 60(4) Quotients 64(2) Composition of Series 66(2) 4. Analytic Functions 68(4) 5. Differentiation of Power Series 72(4) 6. The Inverse and Open Mapping Theorems 76(7) 7. The Local Maximum Modulus Principle 83(3) CHAPTER III Cauchy's Theorem, First Part 86(47) 1. Holomorphic Functions on Connected Sets 86(8) Appendix: Connectedness 92(2) 2. Integrals Over Paths 94(10) 3. Local Primitive for a Holomorphic Function 104(6) 4. Another Description of the Integral Along a Path 110(5) 5. The Homotopy Form of Cauchy's Theorem 115(4) 6. Existence of Global Primitives. Definition of the Logarithm 119(6) 7. The Local Cauchy Formula 125(8) CHAPTER IV Winding Numbers and Cauchy's Theorem 133(23) 1. The Winding Number 134(4) 2. The Global Cauchy Theorem 138(11) Dixon's Proof of Theorem 2.5 (Cauchy's Formula) 147(2) 3. Artin's Proof 149(7) CHAPTER V Applications of Cauchy's Integral Formula 156(17) 1. Uniform Limits of Analytic Functions 156(5) 2. Laurent Series 161(4) 3. Isolated Singularities 165(8) Removable Singularities 165(1) Poles 166(2) Essential Singularities 168(5) CHAPTER VI Calculus of Residues 173(35) 1. The Residue Formula 173(18) Residues of Differentials 184(7) 2. Evaluation of Definite Integrals 191(17) Fourier Transforms 194(3) Trigonometric Integrals 197(2) Mellin Transforms 199(9) CHAPTER VII Conformal Mappings 208(33) 1. Schwarz Lemma 210(2) 2. Analytic Automorphisms of the Disc 212(3) 3. The Upper Half Plane 215(5) 4. Other Examples 220(11) 5. Fractional Linear Transformations 231(10) CHAPTER VIII Harmonic Functions 241(50) 1. Definition 241(11) Application: Perpendicularity 246(2) Application: Flow Lines 248(4) 2. Examples 252(7) 3. Basic Properties of Harmonic Functions 259(12) 4. The Poisson Formula 271(5) The Poisson Integral as a Convolution 273(3) 5. Construction of Harmonic Functions 276(10) 6. Appendix. Differentiating Under the Integral Sign 286(5) PART TWO Geometric Function Theory 291(46) CHAPTER IX Schwarz Reflection 293(13) 1. Schwarz Reflection (by Complex Conjugation) 293(4) 2. Reflection Across Analytic Arcs 297(6) 3. Application of Schwarz Reflection 303(3) CHAPTER X The Riemann Mapping Theorem 306(16) 1. Statement of the Theorem 306(2) 2. Compact Sets in Function Spaces 308(3) 3. Proof of the Riemann Mapping Theorem 311(3) 4. Behavior at the Boundary 314(8) CHAPTER XI Analytic Continuation Along Curves 322(15) 1. Continuation Along a Curve 322(9) 2. The Dilogarithm 331(4) 3. Application to Picard's Theorem 335(2) PART THREE Various Analytic Topics 337(142) CHAPTER XII Applications of the Maximum Modulus Principle and Jensen's Formula 339(33) 1. Jensen's Formula 340(6) 2. The Picard-Borel Theorem 346(8) 3. Bounds by the Real Part, Borel-Caratheodory Theorem 354(2) 4. The Use of Three Circles and the Effect of Small Derivatives 356(4) Hermite Interpolation Formula 358(2) 5. Entire Functions with Rational Values 360(5) 6. The Phragmen-Lindelof and Hadamard Theorems 365(7) CHAPTER XIII Entire and Meromorphic Functions 372(19) 1. Infinite Products 372(4) 2. Weierstrass Products 376(6) 3. Functions of Finite Order 382(5) 4. Meromorphic Functions, Mittag-Leffer Theorem 387(4) CHAPTER XIV Elliptic Functions 391(17) 1. The Liouville Theorems 391(4) 2. The Weierstrass Function 395(5) 3. The Addition Theorem 400(3) 4. The Sigma and Zeta Functions 403(5) CHAPTER XV The Gamma and Zeta Functions 408(32) 1. The Differentiation Lemma 409(4) 2. The Gamma Function 413(18) Weierstrass Products 413(3) The Gauss Multiplication Formula (Distribution Relation) 416(2) The (Other) Gauss Formula 418(2) The Mellin Transform 420(2) The Stirling Formula 422(2) Proof of Stirling's Formula 424(7) 3. The Lerch Formula 431(2) 4. Zeta Functions 433(7) CHAPTER XVI The Prime Number Theorem 440(13) 1. Basic Analytic Properties of the Zeta Function 441(5) 2. The Main Lemma and its Application 446(3) 3. Proof of the Main Lemma 449(4) Appendix 453(26) 1. Summation by Parts and Non-Absolute Convergence 453(4) 2. Difference Equations 457(4) 3. Analytic Differential Equations 461(4) 4. Fixed Points of a Fractional Linear Transformation 465(2) 5. Cauchy's Formula for C(XXX) Functions 467(5) 6. Cauchy's Theorem for Locally Integrable Vector Fields 472(7) Bibliography 479(2) Index 481